U.S. patent application number 10/846281 was filed with the patent office on 2005-01-27 for loop filter for class d amplifiers.
This patent application is currently assigned to Texas Instruments Incorporated. Invention is credited to Neesgaard, Claus N., Risbo, Lars.
Application Number | 20050017799 10/846281 |
Document ID | / |
Family ID | 34083106 |
Filed Date | 2005-01-27 |
United States Patent
Application |
20050017799 |
Kind Code |
A1 |
Risbo, Lars ; et
al. |
January 27, 2005 |
Loop filter for class D amplifiers
Abstract
A class-D amplifier circuit (30; 30') providing improved
open-loop error for base-band frequencies, such as the audio band,
is disclosed. The amplifier circuit (30; 30') includes a comparator
(35) for generating a pulse-width-modulated output signal that is
applied to an output power stage (37). An LC filter (32) is at the
output of the power stage (37). The amplifier circuit (30; 30')
includes a loop filter having multiple feedback loop paths, with at
least one feedback loop path coupled to the output of the power
stage (37), and optionally, at least one feedback loop path coupled
to the output of the LC filter (32). The transfer function
(H.sub.mae(s)) of the loop filter has a real part that has a much
steeper slope (on the order of 80 dB/decade) at frequencies above
the pulse-width-modulation switching frequency than the slope of
its magnitude characteristic at frequencies below this switching
frequency.
Inventors: |
Risbo, Lars; (Copenhagen,
DK) ; Neesgaard, Claus N.; (Plano, TX) |
Correspondence
Address: |
TEXAS INSTRUMENTS INCORPORATED
P O BOX 655474, M/S 3999
DALLAS
TX
75265
|
Assignee: |
Texas Instruments
Incorporated
Dallas
TX
|
Family ID: |
34083106 |
Appl. No.: |
10/846281 |
Filed: |
May 14, 2004 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
60471498 |
May 16, 2003 |
|
|
|
Current U.S.
Class: |
330/10 |
Current CPC
Class: |
H03F 3/217 20130101;
H03F 3/2171 20130101; H03F 2200/331 20130101; H03F 2200/78
20130101 |
Class at
Publication: |
330/010 |
International
Class: |
H03F 003/38 |
Claims
What is claimed is:
1. A class D amplifier circuit, comprising: an adder, for
generating a difference signal responsive to an input signal and a
feedback signal; a pulse-width-modulator, for comparing the
difference signal to a waveform having a switching frequency, and
for generating a pulse-width-modulated output signal responsive to
the result of the comparing; a power stage, for driving its output
responsive to the pulse-width-modulated output signal; and a loop
filter, coupled to the output of the power stage, for generating
the feedback signal by applying a minimum-aliasing-error transfer
function to a signal from the output of the power stage, the
minimum-aliasing-error transfer function being a minimum of second
order, and having a time-domain impulse response that has a
time-domain impulse response h(t) having a slope that is
substantially zero for small positive values of t.
2. The circuit of claim 1, wherein the loop filter comprises a
plurality of feedback loop paths, each having a component transfer
function.
3. The circuit of claim 2, wherein the component transfer function
of at least one of the plurality of feedback loop paths is of at
least second-order.
4. The circuit of claim 2, further comprising: an LC filter,
coupled to the output of the power stage, the LC filter having a
transfer function, and having an output; wherein at least one of
the plurality of feedback loop paths is coupled to the output of
the power stage; and wherein at least one of the plurality of
feedback loop paths is coupled to the output of the LC filter.
5. The circuit of claim 4, wherein the at least one of the
plurality of feedback loop paths coupled to the output of the power
stage has a component transfer function of first order.
6. The circuit of claim 4, wherein the transfer function of the LC
filter is of at least second order.
7. The circuit of claim 1, wherein the minimum-aliasing-error
transfer function is at least third order.
8. The circuit of claim 1, wherein the minimum-aliasing-error
transfer function has a magnitude characteristic versus frequency
that has a most negative slope for frequencies below the switching
frequency that is flatter than the most negative slope of the real
part of the minimum-aliasing-error transfer function at frequencies
above the switching frequency.
9. The circuit of claim 8, wherein the most negative slope of the
magnitude characteristic versus frequency for frequencies below the
switching frequency is about 40 dB/decade; and wherein the most
negative slope of the real part of the minimum-aliasing-error
transfer function at frequencies above the switching frequency is
about 80 dB/decade.
10. A class D amplifier circuit, comprising: an adder, for
generating a difference signal responsive to an input signal and a
feedback signal; a pulse-width-modulator, for comparing the
difference signal to a waveform having a switching frequency, and
for generating a pulse-width-modulated output signal responsive to
the result of the comparing; a power stage, for driving its output
responsive to the pulse-width-modulated output signal; and a loop
filter, coupled to the output of the power stage, for generating
the feedback signal by applying a minimum-aliasing-error transfer
function to a signal from the output of the power stage, the
minimum-aliasing-error transfer function being a minimum of second
order, and having a magnitude characteristic versus frequency that
has a most negative slope for frequencies below the switching
frequency that is flatter than the most negative slope of the real
part of the minimum-aliasing-error transfer function at frequencies
above the switching frequency.
11. The circuit of claim 10, wherein the loop filter comprises a
plurality of feedback loop paths, each having a component transfer
function.
12. The circuit of claim 11, wherein the component transfer
function of at least one of the plurality of feedback loop paths is
of at least second-order.
13. The circuit of claim 11, further comprising: an LC filter,
coupled to the output of the power stage, the LC filter having a
transfer function, and having an output; wherein at least one of
the plurality of feedback loop paths is coupled to the output of
the power stage; and wherein at least one of the plurality of
feedback loop paths is coupled to the output of the LC filter.
14. The circuit of claim 13, wherein the at least one of the
plurality of feedback loop paths coupled to the output of the power
stage has a component transfer function of first order.
15. The circuit of claim 13, wherein the transfer function of the
LC filter is of at least second order.
16. The circuit of claim 10, wherein the most negative slope of the
magnitude characteristic versus frequency for frequencies below the
switching frequency is about 40 dB/decade; and wherein the most
negative slope of the real part of the minimum-aliasing-error
transfer function at frequencies above the switching frequency is
about 80 dB/decade.
17. The circuit of claim 10, wherein the minimum-aliasing-error
transfer function is at least third order.
18. The circuit of claim 10, wherein the loop filter has a
time-domain impulse response h(t) having a slope that is
substantially zero for small positive values of t.
19. A class D amplifier circuit, comprising: a
pulse-width-modulator, for comparing a difference signal to a
waveform having a switching frequency, and for generating a
pulse-width-modulated output signal responsive to the result of the
comparing; a power stage, for driving its output responsive to the
pulse-width-modulated output signal; an LC filter, coupled to the
output of the power stage, the LC filter having a transfer
function, and having an output; and a loop filter, comprising: at
least a first feedback loop path, coupled to the output of the
power stage and producing a first feedback signal, and having a
first order transfer function; a second feedback loop path and
producing a second feedback signal, coupled to the output of the LC
filter; and a multiplicative filter function, for applying a
transfer function of at least first order to a sum including the
first and second feedback signals and an input signal, to produce
the difference signal.
20. The circuit of claim 19, wherein the loop filter further
comprises: a third feedback loop path, coupled to the output of the
power stage and producing a third feedback signal, and having a
first order transfer function. wherein the multiplicative filter
function applies its transfer function to a sum including the
first, second, and third feedback signals and an input signal.
21. The circuit of claim 20, wherein the loop filter further
comprises: first, second, and third gain stages, for applying
first, second, and third gain values, respectively, in generating
the first, second, and third feedback signals.
22. The circuit of claim 20, wherein the multiplicative filter
function comprises: an operational amplifier, having an input
coupled to an adder node, and having an output coupled to the
pulse-width-modulator to present the difference signal; wherein the
first feedback loop path comprises a first R-C network connected
between the output of the power stage and the adder node; wherein
the third feedback loop path comprises a second first R-C network
connected between the output of the power stage and the adder node;
wherein the second feedback loop path comprises a resistor network
connected between the output of the LC filter and the adder node;
and wherein the adder node also receives the input signal.
23. The circuit of claim 19, wherein the loop filter has a transfer
function that has a magnitude characteristic versus frequency that
has a most negative slope for frequencies below the switching
frequency that is flatter than the most negative slope of the real
part of the minimum-aliasing-error transfer function at frequencies
above the switching frequency.
24. The circuit of claim 23, wherein the most negative slope of the
magnitude characteristic versus frequency for frequencies below the
switching frequency is about 40 dB/decade; and wherein the most
negative slope of the real part of the minimum-aliasing-error
transfer function at frequencies above the switching frequency is
about 80 dB/decade.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application claims priority, under 35 U.S.C.
.sctn.119(e), of U.S. Provisional Application No. 60/471,498, filed
May 16, 2003.
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT
[0002] Not applicable.
BACKGROUND OF THE INVENTION
[0003] This invention is in the field of audio amplifiers, and is
more specifically directed to pulse-width modulated class D audio
power amplifiers.
[0004] As is fundamental in the art, electronic amplifier circuits
are often classified in various "classes". For example, the output
drive transistors of class A amplifier circuits conduct DC current
even with no audio signal, and the entire output voltage swing is
of a single polarity. A class B amplifier, on the other hand,
typically includes complementary output drive transistors, driving
an output voltage swing including both positive and negative
polarity excursions. Class B amplifiers are necessarily more
efficient, because both complementary output drive transistors are
never on at the same time. Class AB amplifiers maintain a small
bias current through complementary output drive transistors, so
that the output voltage swing is centered slightly above (or below)
ground voltage. While the non-zero bias current renders class AB
amplifiers theoretically less efficient than class B amplifiers,
class AB amplifiers avoid the crossover distortion of class B
amplifiers.
[0005] In recent years, digital signal processing techniques have
become prevalent in many electronic systems. The fidelity provided
by digital techniques has increased dramatically with the switching
speed of digital circuits. In audio applications, the switching
rates of modem digital signal processing are sufficiently fast that
digital techniques have become accepted for audio electronic
applications, even by many of the fussiest "audiophiles".
[0006] Digital techniques for audio signal processing now extend to
the driving of the audio output amplifiers. A new class of
amplifier circuits has now become popular in many audio
applications, namely "class D" amplifiers. Class D amplifiers drive
a complementary output signal that is digital in nature, with the
output voltage swinging fully from "rail-to-rail" at a duty cycle
that varies with the audio information. Complementary
metal-oxide-semiconductor (CMOS) output drive transistors are thus
suitable for class D amplifiers, as such devices are capable of
high, full-rail, switching rates such as desired for digital
applications. As known in the art, CMOS drivers conduct extremely
low DC current, and their resulting efficiency is especially
beneficial in portable and automotive audio applications, and also
small form factor systems such as flat-panel LCD or plasma
televisions. In addition, the ability to realize the audio output
amplifier in CMOS enables integration of an audio output amplifier
with other circuitry in the audio system, further improving
efficiency and also reducing manufacturing cost of the system. This
integration also provides performance benefits resulting from close
device matching between the output devices and the upstream
circuits, and from reduced signal attenuation.
[0007] In addition to audio amplifiers, class D amplifiers are also
now used in other applications. Modern switching power supplies
utilize class D power amplifier techniques. Class D amplifiers are
also used in some motor control applications, such as voice coil
motors for positioning the read/write heads in modem computer disk
drives.
[0008] By way of background, FIG. 1 illustrates a basic
conventional natural sampling pulse width modulator 1, in an
open-loop mode, as used to generate a pulse width modulated (PWM)
output signal. As shown in FIG. 1, conventional pulse width
modulator 1 includes comparator 5, which compares an input signal
x(t) at its positive input with a unity amplitude triangle wave,
generated by signal source 3 and appearing at its negative input,
to produce a two-level PWM output signal p(t). The triangle
waveform is at a period T and a switching frequency F.sub.sw, as
shown. In this example, output signal p(t) is at an amplitude of +1
responsive to input signal x(t) being instantaneously higher than
the current state of the triangle waveform, and at an amplitude of
-1 responsive to input signal x(t) instantaneously being lower than
the current state of the triangle waveform. In this unity gain
example, if input signal x(t) is at a DC level k, the mean value of
PWM output signal p(t) over time is also at a level k.
[0009] In this conventional natural sampling PWM modulator 1 for AC
input signals x(t) at an input frequency F.sub.in, the modulation
is theoretically perfectly linear, in the sense that no harmonic
distortion is produced by comparator 5. However, non-harmonic
components are produced, at side bands defined by the signal input
frequency F.sub.in, corresponding to multiples of the switching
carrier frequency F.sub.sw:
N.multidot.F.sub.sw.+-.M.multidot.F.sub.in (1)
[0010] These non-harmonic components are minimized if the switching
(i.e., carrier) frequency F.sub.sw is significantly higher than the
input frequency of interest F.sub.in. In audio applications, this
situation is typically present.
[0011] In practice, however, non-idealities in the observed
electrical performance of conventional natural sampling PWM
modulator 1 indicate deviations from theoretical behavior,
especially from the downstream switching power stage that is
controlled by PWM output signal p(t). For example, noise and
distortion arises from switching delays in the downstream power
stage that vary non-linearly with load current. In the modulator
itself, errors such as amplitude distortion and noise in the
triangle wave signal will be evident as distortion and noise in the
PWM output signal p(t). Comparator 5 may itself also contribute to
distortion and noise. In addition, noise, ripple, and variations in
the power supply voltage biasing the downstream switching stage
will also introduce errors in the ultimate output.
[0012] According to conventional approaches, feedback control
compensates for many of these non-ideal effects. FIG. 2 illustrates
a conventional arrangement for a feedback-controlled PWM modulator
1. In FIG. 2, output power stage 7 is shown, as receiving PWM
output signal p(t) and driving the ultimate output signal y(t) for
driving audio speakers or the like. In this example, input signal
x(t) to modulator 1 is derived from ultimate input signal i(t)
combined with a feedback signal from output signal y(t). Output
signal y(t) is subtracted from input signal i(t) by the operation
of inverter 9 and adder 11. The difference signal from adder 11 is
applied to loop filter 13, which produces modulator input signal
x(t) after application of transfer function H(s). Transfer function
H(s) determines both the stability of the system, and the extent to
which error is suppressed by the feedback loop.
[0013] The system of FIG. 2 can be analyzed by considering it as a
linear system with an additional input d(t) that represents the
system error from all causes. This model is illustrated in FIG. 3,
in which modulator 1 and power stage 7 are represented by linear
gain stage 17. Adder 15 applies modeled error input d(t) to the
output of gain stage 17. In the case of FIG. 2, in which the
triangle wave amplitude and the power supply voltage are both
unity, gain K is also unity (assuming a constant power supply
voltage). One can characterize the error transfer function ETF(s)
as follows: 1 ETF ( s ) = 1 1 + K H ( s ) ( 2 )
[0014] where K is the gain applied by gain stage 17. This error
transfer function ETF(s) is the transfer function of error signal
d(t) as it affects output signal p(t). The stability of the overall
system can be determined from the poles of error transfer function
ETF(s), and as such this stability depends on the gain K (which
depends upon the power supply voltage) and on the transfer function
H(s) of loop filter 13. Error suppression can be maximized by
maximizing the gain of the loop filter 13 at the frequencies of
interest; as evident from equation (2), the error suppression
(i.e., the reciprocal of error transfer function ETF(s)) is
effectively the loop filter gain itself, when this gain is
sufficiently high.
[0015] The signal transfer function STF(s): 2 STF ( s ) = K H ( s )
1 + K H ( s ) ( 3 )
[0016] is substantially at unity gain in the band of interest
(i.e., the frequencies at which the gain of loop filter 13 is
high).
[0017] For the sake of this discussion, the system can be
normalized so that gain K is unity, for example by normalizing the
transfer function H(s) of loop filter 13 with the gain of modulator
1 and power stage 7, and by including any scaling in the feedback
path. In effect, all gains outside of loop filter 13 can be
considered as moved into, and thus compensated by, transfer
function H(s). As typical in the art, the description in this
specification will assume such normalization for clarity of
description, although it is to be understood that gain values
outside of the loop could be at values other than unity if
desired.
[0018] Another non-ideal factor that affects the fidelity of class
D amplifiers is ripple in the output signal p(t). More
specifically, stability is optimized by the switching frequency of
the PWM output signal being equal to the switching frequency
F.sub.sw of the triangular waveform. This characteristic is ensured
by limiting the slew rate of the output of loop filter 13 to no
more than the slew rate of the triangular waveform, which prevents
the race-around condition in which the output of comparator 5
oscillates multiple times within a single period of the triangular
waveform; this slew rate limitation holds true so long as waveform
generator 3 in modulator 1 generates a substantially perfect
triangle wave. These conditions also place an additional constraint
on the transfer function H(s) of loop filter 13. It can readily be
derived that ripple stability is attained by constraining the
amplitude gain of transfer function H(s) at switching frequency
F.sub.sw: 3 H ( F sw ) 1 ( 4 )
[0019] Conventional loop filters 13 typically have a slope of
around 20 dB/decade at and just below the switching frequency, in
order to ensure loop stability (i.e., placing closed loop poles in
the left-hand plane). This constrains the unity gain frequency
F.sub.unity to: 4 F unity F sw ( 5 )
[0020] FIG. 4 illustrates a typical log-log response plot for a
conventional loop filter in a natural sampling PWM modulator such
as that in FIG. 2. At lower frequencies, the response slope is
second-order, so that the error suppression carried out by the loop
is maximized, while at higher frequencies, there are one or more
zeros that reduces the slope to first order. In this regard, the
unity gain frequency F.sub.unity is less than the switching
frequency F.sub.sw ensuring ripple stability as described above.
FIG. 4 also illustrates that the maximum loop gain (i.e., the error
suppression) at frequency F.sub.audio (the upper limit of the audio
band) is a function of the ratio of switching frequency F.sub.sw to
audio frequency F.sub.audio. In general, a loop filter may have a
magnitude characteristic over frequency with slopes that are higher
than second-order, provided that there are zeroes that reduce the
slope to near first-order (6 dB/octave) around the unity gain
frequency F.sub.unity. These higher order loop filters will provide
higher error suppression in the audio band.
[0021] Another concern faced by the designer of a PWM loop for
audio amplification is the error due to aliasing in the feedback
loop. As evident from this description, two PWM transitions occur
in each switching period T, so that the sampling frequency is
2F.sub.sw. If the input signal x(t) has frequency components above
the Nyquist frequency (F.sub.sw), aliasing will be present in the
output. More specifically, those components in input signal x(t)
that are at frequencies near the switching frequency F.sub.sw will
appear into the lower frequency audio band. This aliasing is, of
course, undesirable for audio fidelity.
[0022] Referring back to FIG. 2, it is seen that high frequency
components of the PWM output signal p(t) will feed back to the
input of comparator 5. These high components are referred to as the
ripple signal, which will produce an aliasing error when sampled by
the comparator. In addition, because PWM output signal p(t) will
have multiple side bands for each harmonic of switching frequency
F.sub.sw, these side bands will also alias down as harmonic
distortion.
[0023] By way of further background, Berkhout, "Integrated Class D
Amplifier", presented at the 112th Conventional of the Audio
Engineering Society (May 10-13, 2002; Munich), describes a class D
amplifier that includes a second-order loop filter 13. The transfer
function H(s) for this conventional filter is a weighted sum of a
first order integrator and a second order integrator: 5 H ( s ) = K
1 s + K 2 s 2 = K 1 s + K 2 s 2 ( 6 )
[0024] The first order integrator is a typical loop filter transfer
function for simple class D amplifiers, as it has a zero real part
for all positive integer multiples of the switching frequency
F.sub.sw, and thus produces no aliasing DC error. As known in the
art, the second order integrator increases the error suppression in
the base-band. The summed first and second order terms in the
Berkhout loop filter introduces a real zero at s=-K.sub.2/K.sub.1
that reduces the phase characteristic to 90.degree. for loop
stability. However, it has been observed that the error function of
the loop will be effectively set by the second-order integrator,
and the error will scale with K.sub.2. The error suppression in the
base-band also scales with K.sub.2, in which case the closed loop
error will effectively be constant, such that increasing the
second-order feedback by scaling K.sub.2 will be ineffective in
decreasing distortion.
BRIEF SUMMARY OF THE INVENTION
[0025] It is therefore an object of this invention to provide a
class D amplifier having a loop filter in which the aliasing error
is minimized in the base-band frequencies of interest.
[0026] It is a further object of this invention to provide such a
class D amplifier in which the stability of the loop filter is not
decreased while reducing the aliasing error.
[0027] It is a further object of this invention to provide such a
class D amplifier in which the amplitude characteristic has a
sharper characteristic at low frequencies.
[0028] It is a further object of this invention to provide such an
amplifier in which error inserted at the output LC filter is also
included in the feedback loop.
[0029] It is a further object of this invention to provide such an
amplifier in which the effects of load resistance are compensated
by the loop filter.
[0030] It is a further object of this invention to provide such an
amplifier in which these benefits are attained with minimum
complexity in the feedback loop filter realization.
[0031] Other objects and advantages of this invention will be
apparent to those of ordinary skill in the art having reference to
the following specification together with its drawings.
[0032] The present invention may be implemented into a class D
amplifier in which a feedback loop filter is provided. The loop
filter includes at least one loop path with a transfer function of
second order or higher, and at least one loop path in which the
maximum negative slope of the magnitude versus frequency for
frequencies below the switching frequency is lower than the
negative slope of its real part at frequencies above the switching
frequency. The loop filter can be realized as a single loop filter,
in which the multiple loop paths operate from a feedback signal at
the output of the power stage.
[0033] According to another aspect of the invention, the loop
filter is implemented in a dual loop feedback arrangement, taking
feedback signals on each side of an LC filter at the output of the
class D amplifier. The LC filter characteristic not only filters
the output signal of the amplifier, but also implements a second
order loop path in the feedback loop filter that is summed with
first order loop paths.
BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING
[0034] FIG. 1 is an electrical diagram, in block form, of a
conventional pulse-width-modulator.
[0035] FIG. 2 is an electrical diagram, in block form, of a
conventional class D amplifier incorporating a
pulse-width-modulator driving an output stage, and a feedback loop
filter.
[0036] FIG. 3 is an electrical diagram, in block form, modeling the
conventional class D amplifier of FIG. 2.
[0037] FIG. 4 is a plot of the frequency characteristic of a
conventional loop filter.
[0038] FIG. 5a is an electrical diagram illustrating an analytical
model, and
[0039] FIGS. 5b and 5c are timing diagrams illustrating the DC
error caused by ripple in the feedback loop signal in this
analytical model, as observed according to this invention.
[0040] FIG. 6 is an electrical diagram, in block form, of a model
of error in a class D amplifier, as observed according to the
invention.
[0041] FIGS. 7a and 7b are timing diagrams illustrating a period of
a fixed (but arbitrary) duty cycle PWM output signal p(t) and the
corresponding error function Fourier series g.sub.m(y) (i.e., for
m=1, 2, 3, 4), respectively.
[0042] FIG. 8 is an electrical diagram, in block form, of an audio
system constructed according to the preferred embodiments of the
invention.
[0043] FIG. 9 is an electrical diagram, in block form, of the audio
amplifier in the system of FIG. 8 constructed according to the
preferred embodiments of the invention.
[0044] FIG. 10 is a frequency response plot of a loop filter
according to the preferred embodiments of the invention.
[0045] FIG. 11 is a frequency response plot illustrating the
magnitude and real parts of a loop filter according to the
preferred embodiments of the invention as compared with a
conventional loop filter.
[0046] FIG. 12 is a plot of the phase response of a loop filter
according to the preferred embodiments of the invention as compared
with a conventional loop filter.
[0047] FIG. 13 is a plot of the time-domain impulse response of a
loop filter according to the preferred embodiments of the invention
as compared with a conventional loop filter and with a first-order
integrator.
[0048] FIG. 14 is a plot of the aliasing error function of a loop
filter according to the preferred embodiments of the invention as
compared with a conventional loop filter.
[0049] FIG. 15 is a plot of the minimum damping factor of a loop
filter according to the preferred embodiments of the invention as
compared with a conventional loop filter.
[0050] FIG. 16 is an electrical diagram, in block form, of an audio
amplifier according to a second preferred embodiment of the
invention.
[0051] FIG. 17 is an electrical diagram, in schematic form, of an
exemplary realization of the feedback loop of the audio amplifier
of FIG. 16 according to the second preferred embodiment of the
invention.
DETAILED DESCRIPTION OF THE INVENTION
[0052] The present invention will be described in connection with
its preferred embodiment, namely as implemented into an audio
amplifier, because it is contemplated that this invention is
especially beneficial in such an application. However, it is also
contemplated that the benefits of this invention can also be
attained in other applications, such as switching power supplies,
and motor control drivers such as used in disk drives. Accordingly,
it is to be understood that the following description is provided
by way of example only, and is not intended to limit the true scope
of this invention as claimed.
[0053] The present invention is based on certain observations
regarding the operation of loop filters in class D amplifiers, such
as the conventional feedback-controlled class D amplifier of FIG.
2. As mentioned above, high frequency components of the output
signal p(t) from conventional PWM modulators will feed back to the
input of the comparator of the PWM modulator itself. These high
frequency components are referred to as the ripple signal, r(t),
which produces an aliasing error when sampled by the comparator. In
addition, because PWM output signal p(t) will have multiple side
bands for each harmonic of switching frequency Flo these side bands
will also alias down as harmonic distortion.
[0054] It has been discovered, according to this invention, that
the ripple signal r(t) will introduce a DC error in the output
signal. It was discovered, in connection with this invention, that
there is not a closed-loop form that expresses this error as a
function of the input signal. In connection with this invention, an
analytical model of the open-loop operation of a class D audio
amplifier, as illustrated in FIG. 5a, was used to derive this
relationship. In this model, a DC input level x is applied to PWM
modulator 1, which generates an output PWM signal p'(t). This
output PWM signal p'(t) is not used in the feedback loop, however.
Rather, pulse generator 27 is provided to generate a feedback pulse
p(t) having a mean value y, with pulse p(t) applied to ideal
high-pass filter 29, and in turn to loop filter 13. The goal of the
analytical model is to find a DC input level x that, when applied
to PWM modulator 1, produces an output PWM signal p'(t) that is
exactly equal to the output p(t) from pulse generator 27. The
difference between input level x and the mean level y of the output
p(t) constitutes the open-loop DC error.
[0055] For a perfectly linear PWM modulator, one would expect that
input level x would exactly equal mean level y. However, ripple
signal r(t) from loop filter 13 is added to the DC input level x,
and introduces DC error. It is this open-loop DC error G(x)=x-y as
a function of input level x that is sought. In FIG. 5b, the signal
x+r(t) corresponds to the DC input signal x plus this ripple signal
r(t), which is in the form of a periodic waveform of the same
frequency as the switching frequency F.sub.sw (due to conceptual
high-pass filter 29 in the loop). FIG. 5b illustrates the output
mean value y, and also the crossing points A, B between the input
signal and the triangle waveform that would generate this output
mean value y. Because of ripple signal r(t), however, crossing
points C and D are the actual crossing points of the input signal
x+r(t) with the triangle waveform that provide the output mean
value that will result in an output PWM signal p(t)=p'(t) having
the output mean value y. The input level x required to place the
model circuit of FIG. 5a in this state (i.e., p(t)=p'(t)) is shown
in FIG. 5b. The open-loop DC error G(x)=y-x is apparent from FIG.
5b, as is the corresponding phase error corresponding to delay time
T.sub.D.
[0056] FIG. 5c zooms into the portion of the triangle waveform
between about crossing point C and crossing point D. Crossing point
C corresponds to the amplitude x+R.sub.r of the input signal when
it crosses the triangle waveform at time T.sub.r, and crossing
point D corresponds to the amplitude x+R.sub.f of the input signal
when it crosses the triangle waveform at time T.sub.f, as shown.
Delay time T.sub.D is the point in time at which the triangle
waveform reaches its minimum amplitude (-1), as shown in FIG. 5b.
One can derive the phase error as a function of this frequency,
considering that times T.sub.r and T.sub.f can be expressed as: 6 T
r = - y + 1 4 F sw ( 7 a ) T f = y + 1 4 F sw ( 7 b )
[0057] while the ripple values R.sub.r, R.sub.f correspond to:
R.sub.r=r(T.sub.r) (8a)
R.sub.f=r(T.sub.f) (8b)
[0058] From FIG. 5b, one can see that the slope of the triangle
wave is 4F.sub.sw. Accordingly, one can readily derive:
(T.sub.f-T.sub.D)4F.sub.sw=x+R.sub.f+1 (9a)
[0059] which can be rearranged to: 7 T D = T f - x + R f + 1 4 F sw
( 9 b ) Similarly : ( T D - T r ) 4 F sw = x + R r + 1 ( 9 c ) and
: T D = T r + x + R r + 1 4 F sw ( 9 d )
[0060] The combination of equations (9b) and (9d) provides: 8 x = 2
F sw ( T r - T D ) - 1 - R r + R f 2 ( 10 a ) and thus : x = y - R
r + R f 2 ( 10 b )
[0061] Equation (10b) thus provides an explicit expression for the
DC error y-x that results from the ripple signal, from which one
can derive the amplitude error function as: 9 G ( y ) = y - x = R r
+ R f 2 ( 11 )
[0062] Referring now to FIG. 6, it has been discovered, in
connection with this invention, that one can consider the entire
closed-loop system of FIGS. 2 and 3 as a base-band model in which
the PWM modulator and output stage 21 is modeled by an
amplitude-error-function 19 having a transfer function G(y), which
derives the error signal d(t) from the output signal y(t) that is
added to the modulator input signal x(t) by adder 23. In the
base-band, which includes low frequencies including the frequencies
of interest (e.g., in the audio band) and extending to a certain
fraction of the switching frequency F.sub.sw, analogous to a system
having an ideal low pass filter that blocks out all switching
related frequency components. Amplitude-error-function 19 of FIG. 6
models the DC aliasing error by a function G(y) that is accurate
over these low frequencies, and that will appear as harmonic
distortion for a sine wave input x(t). However, as shown by this
model, the error function G(y) is introduced inside of the feedback
loop, and will therefore be suppressed by the loop filter gain (or,
stated another way, shaped by the error transfer function
ETF(s)).
[0063] As a result, it has been observed, according to this
invention, that the harmonic distortion of the system of FIG. 6 is
solely a function of loop filter 13 and its transfer function H(s).
It has been discovered, according to this invention, that the total
harmonic distortion in the system can be reduced by increasing the
base-band loop gain in transfer function H(s) to increase
suppression of the harmonic distortion at those frequencies, and by
optimizing the loop filter so that the open-loop error G(y) is
minimized. According to the preferred embodiments of the invention,
as will be described below, both of these approaches are used to
reduce total harmonic distortion.
[0064] As described above, ripple signal r(t) is the steady-state
response of the loop filter 13, and its transfer function H(s), to
a periodic pulse train p(t) that is at switching frequency
F.sub.sw. It is instructive to express ripple signal r(t) using
Fourier series techniques. FIG. 7a illustrates one period of a
fixed (but arbitrary) duty cycle PWM output signal p(t). The
Fourier transform G.sub.p(f) of this single-period signal (i.e, all
values outside of the illustrated period are zero) is: 10 G p ( f )
= ( y + 1 F sw ) sin ( f y + 1 2 F sw ) f y + 1 2 F sw - 1 F sw sin
( f F sw ) f F sw ( 12 )
[0065] Because the PWM signal p(t) is effectively the single-period
signal of FIG. 8a repeated at switching frequency F.sub.sw, the
repeated signal can be characterized by a Fourier series based on
samples of the Fourier transform of equation (12) at multiples of
switching frequency F.sub.sw:
G(m)=F.sub.sw.multidot.G.sub.p(mF.sub.sw) (13)
[0066] Limiting the value of sinc(0) to unity, the Fourier series
becomes: 11 G m = 2 sin ( m y + 1 2 ) m , for m 0 = y , for m = 0 (
14 )
[0067] Because G.sub.0=y, the mean value of the PWM output signal
p(t) is y, as desired and as described above. Further, the Fourier
series is real-valued, and therefore the time domain signal p(t) is
even (i.e., p(t)=p(-t)). Ripple signal r(t) can be found by summing
this Fourier series G.sub.m, weighted by the loop filter transfer
function -H(s) sampled at the frequencies mF.sub.sw. Leaving out
the DC coefficient to reflect the ideal conceptual high-pass filter
29 mentioned above: 12 r ( t ) = m = 1 .infin. - 2 exp ( m 2 F sw t
) G m H ( m 2 F sw ) ( 15 )
[0068] This expression permits, with reference to equation (11),
the calculation of the error function G(y) from the loop transfer
function H(s). In particular: 13 G ( y ) = R r + R f 2 = r ( - y -
1 4 F sw ) + r ( y - 1 4 F sw ) 2 = m = 1 .infin. - 2 cos ( m ( y +
1 ) 2 ) P m Re { H ( m 2 F sw ) } = m = 1 .infin. - 4 m cos ( m y +
1 2 ) sin ( m y + 1 2 ) Re { H ( m 2 F sw ) } = m = 1 .infin. - 2 m
sin ( m ( y + 1 ) ) Re { H ( m 2 F sw ) } = m = 1 .infin. 2 ( - 1 )
m + 1 sin ( m y ) m Re { H ( m 2 F sw ) } = m = 1 .infin. g m ( y )
h m , where g m ( y ) = 2 ( - 1 ) m + 1 sin ( m y ) m and h m = Re
{ H ( m 2 F sw ) } ( 16 )
[0069] Derivation (16) illustrates that the imaginary part of the
loop filter transfer function H(s) does not contribute at all to
the error function G(y). This can be intuitively seen by
considering ripple signal r(t) as the sum of even and odd
parts:
r(t)=r.sub.odd(t)+r.sub.even(t) (17)
[0070] Odd part r.sub.odd(t) is defined by the sine terms of the
Fourier series for ripple signal r(t), which is defined solely by
the imaginary part of transfer function H(s), because PWM output
signal p(t) is an even function as mentioned above. But equation
(11) establishes that the odd part of ripple signal r(t) will not
contribute at all to the error function G(y). As a consequence,
only even part r.sub.even(t) of ripple signal r(t) win affect error
function G(y). This is used to advantage in the design of the loop
filter in the class D amplifier according to the preferred
embodiments of the invention, as will be described below.
[0071] Analysis of the first four terms of the error function
Fourier series g.sub.m(y) (i.e., for m=1, 2, 3, 4) is illustrated
in FIG. 7b. As shown, the error function G(y) is zero at output
values y=0, y=+1, and y=-1. This means that there is no DC error
for zero, or full scale (positive or negative) output from the
system. This behavior is also used to advantage in the class D
amplifier according to the preferred embodiments of the invention,
as will be described below.
[0072] Referring next to FIG. 8, a system incorporating the class D
amplifier according to the preferred embodiments of the invention
will first be described, to provide context for the invention. FIG.
8 illustrates an audio system that includes audio output amplifier
circuitry according to the preferred embodiment of the invention.
The audio system of FIG. 8 may represent a standalone audio system,
such as an automobile, portable, or bookshelf sound system, or
alternatively may be implemented within an audio-visual system,
such as a television set. It is contemplated that this invention is
applicable to audio systems in any number of applications,
including these types of audio systems, and also in other systems
that provide audio output.
[0073] In the system of FIG. 8, an audio source provides audio
signals to coder/decoder (codec) 26. The audio source may be any
one of a number of conventional sources of digital or analog audio
information, including compact disc (CD) or digital video disk
(DVD) players, a computer forwarding digital audio information such
as in the form of MPEG data, sources of analog audio information
such as from microphones and musical instrument pickups, or audio
signals communicated by conventional broadcast or cable television
sources. The audio signal, if in the digital domain, may be also be
processed by conventional digital signal processing routines,
including filtering and the like, by a digital signal processor
(DSP) such as the 320C5x digital signal processors available from
Texas Instruments Incorporated. Codec 26 is a conventional codec
device, including such functions as analog-to-digital converters
(ADCs) and digital-to-analog converters (DACs), oversampling
digital interpolation filters, sigma-delta modulators, and the
like. An example of a suitable codec 26 is the TLV320AIC23B
high-performance stereo audio codec available from Texas
Instruments Incorporated.
[0074] The output of codec 26 is applied to preamplifier 28. These
signals may be communicated as a real-time differential analog
voltage for each audio channel, or alternatively as a single-ended
signal, depending upon the system application. Preamplifier 28 may
be a conventional audio preamplifier, realized by relatively
low-voltage devices, relative to the class D power amplifier 30.
According to this embodiment of the invention, preamplifier 28
amplifies the codec output, and applies the amplified signal to
class D audio output amplifier 30. Class D audio output amplifier
30, via LC filter 32, drives one or more speakers SPKRS with
pulse-width modulated (PWM) rail-to-rail signals. As shown in FIG.
7, an optional feedback path is provided from the output of LC
filter 32 back into class D audio output amplifier 30; this
feedback arrangement will be described in detail relative to one of
the preferred embodiments of the invention.
[0075] FIG. 9 illustrates the construction of class D audio output
amplifier according to the preferred embodiments of the invention,
in a generalized form. As shown in FIG. 9, PWM modulator 31
includes triangle waveform generator 33 and comparator 35, and
generates PWM output signal p(t) in response to the comparison
between the current level of input signal x(t) and the current
level of the triangle wave from generator 33. PWM output signal
p(t) controls the drive of power transistors in power stage 37, the
output of which is applied, as output signal y(t), to LC filter 32
(as in FIG. 8). The filtered output of LC filter is the ultimate
signal applied to a load, such as audio speakers (FIG. 8).
[0076] According to the preferred embodiments of the invention,
loop filter 33 receives a feedback signal corresponding to the
output signal y(t) from power stage 37. Optionally, as will be
described below, a second feedback signal may also be received from
the output of LC filter 32, such that LC filter 32 is also within
the feedback loop. The output of loop filter 32 is a feedback
signal that is subtracted (by operation of inverter 39 and adder
41) from the input audio signal i(t) that is received from a codec
or other audio source.
[0077] According to the preferred embodiments of the invention, the
transfer function H.sub.mae(s) applied by loop filter 33 is a
minimum-aliasing-error loop filter. This transfer function
H.sub.mae(s) is selected to provide reduced phase (and therefore
increased phase margin), while also having a reduced real part at
multiples of the switching frequency F.sub.sw, as described
above.
[0078] As described above relative to equation (6), the
conventional loop filter includes a weighted sum of first-order and
second-order loops. For purposes of comparison, it is therefore
useful to consider the first-order and second-order integrator
feedback loops of this conventional transfer function as defining a
piece-wise linear characteristic. First, the h.sub.m series of the
second-order loop filter is: 14 h2 m = Re ( K 2 ( 2 m F sw ) 2 ) =
- K 2 4 2 m 2 1 F sw 2 ( 18 )
[0079] Next, the first order one pole filter has the transfer
function: 15 Ha ( s ) = K 1 p s + p ( 19 )
[0080] where p is the pole frequency. This first order filter
provides an h.sub.m series of: 16 ha m = Re ( K 1 p 2 mF sw + p ) =
K 1 p 2 4 2 m 2 F sw 2 + p 2 ( 20 )
[0081] The error function of the summed transfer function (i.e.,
the sum of the first-order and second-order loops) can be minimized
by setting the weights: 17 K 1 p 2 = K 2 or : ( 21 a ) p = K 2 K 1
( 21 b )
[0082] One can evaluate the second-order integrator transfer
function 18 H2 ( s ) = K 2 / s 2
[0083] of the conventional second-order integrator at s=ip: 19 H2 (
s = ip ) = K 2 ( i K 2 K 1 ) 2 = - K 1 ( 22 )
[0084] As a result, the traditional piece-wise linear
approximations of the amplitude responses of the first-order and
second-order loops results in the approximations intersecting at
the real pole p as shown in FIG. 10.
[0085] According to the preferred embodiments of the invention, the
transfer function Ha(s) for loop filter 33 in amplifier 30 of FIG.
9 is: 20 H mae ( s ) = K 1 p s + p + K 2 s 2 = K 1 K 2 K 1 s + K 2
K 1 + K 2 s 2 = K 2 ( s + K 2 K 1 ) + s 2 K 1 K 2 s 2 ( s + K 2 K 1
) ( 23 )
[0086] This example of transfer function H.sub.mae(s) is a
third-order function, with a double pole at zero frequency and a
real pole at -p. The zeros of transfer function H.sub.mae(s) are
complex, and located at 21 s = K 2 3 K 2 2 .
[0087] The plots of the magnitude response and the real part of the
response of this new transfer function H.sub.mae(s) and the
conventional transfer function H(s) described by equations (18)
through (22) are shown in FIG. 11.
[0088] In FIG. 11, plot 42m is the magnitude plot of the
conventional weighted-sum filter transfer function H(s) following
equation (6) for the case of K.sub.1=K.sub.2=1, and plot 42Re is
the plot of the real part of this transfer function H(s). Plot 40m,
on the other hand, is the magnitude plot of the transfer function
H.sub.mae(s) following equation (23) for the case of
K.sub.1=K.sub.2=p=1 for loop filter 33 in amplifier 30 according to
the preferred embodiment of the invention, while plot 40Re plots
the real part of this transfer function H.sub.mae(s). As evident
from FIG. 11, magnitude plots 40m, 42m are quite similar to one
another, with at most a 4 to 5 dB difference at a radian frequency
.omega.=1.
[0089] However, as intended according to the preferred embodiments
of the invention, the real part of transfer function H.sub.mae(s)
for loop filter 33 according to the preferred embodiments of the
invention, as shown by plot 40Re, has a much steeper slope (on the
order of 80 dB/decade) at frequencies above radian frequency
.omega.=p=1 than does the real part of the conventional transfer
function H(s) as shown by plot 42Re (which has a slope of on the
order of 40 dB/decade). This steeper real part for new transfer
function H.sub.mae(s) results in a much lower aliasing error than
the conventional transfer function H(s). Indeed, the reduction in
aliasing error relative to the aliasing error for the conventional
loop filter of equation (6) is roughly about the square of the
ratio of the switching frequency .omega..sub.sw to the pole
frequency .omega.=p.
[0090] This difference in aliasing error is shown in FIG. 14, which
illustrates plots 40G, 42G of the aliasing error function G(y) for
new transfer function H.sub.mae(s) and conventional transfer
function H(s), respectively, for the example of a switching
frequency at .omega..sub.sw=4 rad/sec. The greatly reduced aliasing
error resulting from the new transfer function H.sub.mae(s) is
readily apparent from FIG. 14, with the peak error being only about
0.00235 as compared with a peak error of about 0.04 for the
conventional loop filter. This reduction of aliasing error is in
accordance with the square-law prediction mentioned above, which
predicts an aliasing error reduction by a factor of
(.omega..sub.sw/p).sup.2=16. The peak error of 4% for the
conventional loop filter would be expected to dominate over the
open-loop error in a well-designed switching power stage.
[0091] FIG. 12 illustrates phase plot 42ph for the conventional
loop filter transfer function H(s), and phase plot 40ph for the new
transfer function H.sub.mae(s) according to the preferred
embodiments of the invention. As shown by phase plot 40ph, the
phase characteristic for the new transfer function H.sub.mae(s) is
much sharper in its transition from -180.degree. to -90.degree. in
phase, around pole frequency p, than is the phase characteristic
for conventional transfer function H(s) as shown by phase plot
42ph.
[0092] FIG. 13 illustrates the time-domain impulse response of a
first-order integrator, of the conventional loop filter, and of the
loop filter having transfer function H.sub.mae(s) and constructed
according to the preferred embodiments of the invention. Plot 41t
in FIG. 13 illustrates the well-known step function response of a
conventional first-order integrator to a unit impulse at time t=0.
Plot 42t illustrates the impulse response of the conventional
weighted sum loop filter, for example as described in the Berkhout
article mentioned above. As evident from plot 42t, the impulse
response of this conventional loop filter is effectively the sum of
a first-order integrator (i.e., the step function at time t=0) with
the linear integration of a second-order integrator. In contrast,
plot 40t of the loop filter according to the preferred embodiments
of the invention, with the new transfer function H(s), has an
impulse response that approximates the impulse response of the
first-order integrator of plot 41t at short time intervals
following time t=0. This is evident by the portion of plot 40t with
approximately zero slope shortly following t=0. At longer time
intervals, however, the loop filter according to the preferred
embodiments of the invention has an impulse response that
asymptotically approaches the linear ramp of the second-order
conventional loop filter. FIG. 13 illustrates this by the tangent
of plot 40t asymptotically approaching the linear slope of plot 42t
for the conventional loop filter. In contrast, the conventional
second-order loop filter impulse response has a fixed slope of
unity for t>0. This time-domain behavior of the loop filter
according to the preferred embodiments of the invention, with the
new transfer function H.sub.mae(s), is reflected by its low
allasing error, by approximating the error-free first-order
integrator at short time scales, while having higher loop gain at
low frequencies as evidenced by the second-order integrator
term.
[0093] The minimum-aliasing-error loop filter described above
relative to equation (23) can be further extended to include a
third-order integrator term 22 K 3 / s 3
[0094] without changing the real part of the transfer function
H.sub.mae(s). This provides further improvement in low frequency
loop gain (i.e., reflected at longer time intervals in the impulse
response of FIG. 13), without adding aliasing error. It is further
contemplated that additional correction terms can be added into the
transfer function H.sub.mae(s) that further reduce its real part at
multiples of switching frequency F.sub.sw, for example achieving a
120 dB/decade slope of the real part at higher frequencies.
Integrator terms of even higher order can also be added; in this
regard, it has been observed that all even-order integrator terms
(2.sup.nd, 4.sup.th, 6.sup.th order, etc.) will benefit from
additional correction terms that reduce the real part of the
transfer function. However, the 2.sup.nd order integrator term is
typically the dominant contribution to the real part of the
transfer function and thus requires the most care in selecting the
proper correction terms to reduce the real value.
[0095] As described above, stability of the loop filter
characteristic is important in realizing a high fidelity class D
amplifier. Stability can be analyzed by defining a damping factor
.xi. from the poles of the error transfer function 23 ETF ( s ) = 1
1 + K 0 + H mae ( s )
[0096] as a function of loop gain K.sub.0. For a complex pole-pair
q, this damping factor .xi. becomes: 24 = - Re { q } q ( 24 )
[0097] namely, the ratio of the negative real part of the pole to
its magnitude. Stable systems have a minimum damping factor for all
complex poles that is greater than zero, and a damping factor of
unity indicates that the poles are real and in the left-hand plane
(i.e., critically-damped, or over-damped if the damping factor is
above unity). FIG. 15 shows plots 40S, 42S, which are plots of the
stability factor for the conventional transfer function H(s) and
new transfer function H.sub.mae(s). As shown in FIG. 15, the
conventional loop filter is critically-damped (with a damping
factor .xi. of unity at K.sub.0=4 due to a double real pole), while
the damping factor .xi. for the new transfer function is only 0.37
at this K.sub.0. This relative loss of stability should be taken
into consideration in implementing the loop filter according to the
preferred embodiments of the invention. It has been further
discovered, in connection with this invention, that the stability
of the minimum-aliasing-error loop filter according to the
preferred embodiments of the invention can be further improved
(i.e., damping factor .xi. increased) by adding a first-order
integrator term, which will not change the aliasing error because
of the zero real part of the 1/s first-order integrator transfer
function. In addition, it has been observed that the aliasing error
G(y) due to propagation delays in power output stage 37 and
comparator 35 is substantially linear up to a certain amplitude of
output amplitude y. However, this amplitude limit scales with a
scaling factor k applied to the gain (nominally unity). As such,
the aliasing error due to system delay should be considered as the
gain of the closed loop is designed.
[0098] Ripple instability is also preferably considered in the
design of loop filter 33 according to the preferred embodiments of
the invention, by ensuring that the slope of the ripple signal is
lower than the slope of the triangular waveform, as mentioned
above. It has been discovered, in connection with this invention,
that ripple instability can be assured for a first-order integrator
loop filter 25 K 1 / s
[0099] with: 26 K 1 ( 1 + y ) < 4 S w K 1 4 ( 1 + y ) F sw 2 F
sw ( 25 )
[0100] This is consistent with the conventional rule of thumb that
ripple stability is assured by the unity gain point of the loop
being at least a factor of is below the switching frequency
F.sub.sw; conversely, the magnitude of transfer function
H.sub.mae(s) at switching frequency F.sub.sw should be less than
1/.pi..
[0101] As mentioned above relative to FIG. 9, the preferred
embodiments of the invention provide loop filter 33 that receives a
first feedback input from the output of power stage 37, and
optionally receives a second feedback input from the output of LC
filter 32, such that the characteristic of LC filter 32 itself is
involved in the feedback loop. According to a first preferred
embodiment of the invention, therefore, only the feedback input
from the output of power stage 37 is applied to loop filter 33. In
this event, the desired transfer function H.sub.mae(s) is
implemented entirely within loop filter 33. It is contemplated that
those skilled in the art having reference to this specification
will be readily able to realize this transfer function H.sub.mae(s)
in loop filter 33, for example by a conventional arrangement of
operational amplifiers and passive components.
[0102] According to a second preferred embodiment of the invention,
feedback inputs to loop filter 33 are received both from the output
of power stage 37, and also from the output of LC filter 32. As
mentioned above, this provides the important benefits of involving
LC filter 32 itself in the loop, so that any error due to LC filter
32 itself is compensated in the operation of amplifier 30. In
addition, any error caused by variations in the ultimate load
impedance can similarly be compensated to some extent. Finally, the
transfer function of LC filter 32 itself can be incorporated into
the loop filter transfer function H.sub.mae(s), which can simplify
the realization of loop filter 33.
[0103] FIG. 16 illustrates a class D audio amplifier 30'
constructed according to this second preferred embodiment of the
invention. In this embodiment of the invention, as before, PWM
modulation is performed by comparator 35, which generates PWM
output signal p(t) in response to the comparison between the
current level of input signal x(t) and the current level of the
triangle wave from generator 33. PWM output signal p(t) controls
the drive of power transistors in power stage 37, the output of
which is applied, as output signal y(t), to LC filter 32. The
output of LC filter 32 is forwarded to the ultimate load of
amplifier 30'.
[0104] According to this embodiment of the invention, the loop
filter is realized by three feedback paths, together with filter
function 43 that is applied to the output of adder 41. Adder 41
receives a summed signal from the three feedback paths, and adds
this summed signal to the audio input signal i(t); the output of
adder 41 is forwarded to filter function 43, which in turn drives
signal x(t) to comparator 35.
[0105] A first feedback path in amplifier 30' from the output of
power stage 37 is filtered by filter function 52 with a filter
having a transfer function 27 s / p a s / p a + 1 .
[0106] The output of filter function 52 is applied to gain stage
53, which applies a gain of -K.sub.a. The output of gain stage 53
is applied to an input of adder 54. A second feedback path in
amplifier 30' also receives the output of power stage 37, and
includes filter function 56 with a transfer function 28 1 s / p a +
1 .
[0107] The output of filter function 56 is amplified by gain stage
57, with a gain of -K.sub.c, and the amplified signal is applied to
an input of adder 58.
[0108] According to this second preferred embodiment of the
invention, a third feedback path includes gain stage 59, which
receives the output of LC filter 32 and applies a gain of -K.sub.d.
Because this third feedback path is taken from the output of LC
filter 32, this third path in effect applies the transfer function
H.sub.LC of LC filter 32 to the output signal y(t). As such, the
feedback signal includes this transfer function H.sub.LC, which in
this example is 29 H LC ( s ) = 1 LCs 2 + L R s + 1 .
[0109] After application of the gain of -K.sub.d by gain stage 59,
the resulting signal is applied to a second input of adder 58. The
output of adder 58 is applied to a second input of adder 54, the
output of which is applied as the summed feedback signal to a
second input of adder 41.
[0110] Filter function 56 filters the difference signal from adder
41 with the transfer function 30 1 s / p b + 1 ,
[0111] according to this embodiment of the invention. The output of
filter function 56 is applied to comparator 35, as signal x(t).
[0112] According to this construction of this second preferred
embodiment of the invention, the overall transfer function
H.sub.mae(s) in amplifier 30' is: 31 H mae ( s ) = [ K d 1 LCs 2 +
L R s + 1 + K a s p a s p a + 1 + K c 1 s p c + 1 ] 1 s p b + 1 (
26 )
[0113] According to the preferred embodiments of the invention, the
parameters of the gains K and poles p are optimized to obtain the
properties of the transfer function H.sub.mae(s) described in this
specification. In general, the inner term with the gain K.sub.d
(i.e.,, the term due to LC filter 32) corresponds to a two-pole
low-pass filter, appearing at high frequencies (i.e., at
frequencies above the switching frequency) as a double integrator.
The first-order inner K.sub.a and K.sub.c terms, in combination
with the outer first-order low-pass with pole p.sub.b, serve as
correction terms to reduce the real-part contribution of the double
integrator term (LC filter 32) and to ensure that the total loop
filter magnitude response slope is about 6 dB/octave above the
switching frequency. The combination of filter parameters (K.sub.a,
K.sub.c, K.sub.d, p.sub.a, p.sub.b, p.sub.c and L, C, R) provide
enough degrees of freedom to facilitate the identification of a
solution meeting the requirements described in this specification.
It is contemplated that those skilled in the art having reference
to this specification will be readily able to perform the
appropriate optimization for a particular application and, once the
optimization is accomplished, to select the values of the passive
components L, C, R according to conventional techniques.
[0114] According to the preferred embodiments of the invention, the
transfer function H.sub.mae(s) is optimized to have the desired
properties, namely having an amplitude response that has a maximal
negative slope of magnitude versus frequency, below a switching
frequency F.sub.sw, that is flatter than the negative slope of the
real part of the response versus frequency for frequencies above
switching frequency F.sub.sw. The transfer function H.sub.mae(s)
has a real part with a much steeper slope (on the order of 80
dB/decade) at frequencies above a selected frequency .omega.=1 than
conventional filters, resulting in much reduced aliasing error.
Additionally, as described above, the phase characteristic for the
new transfer function H.sub.mae(s) is much sharper in its
transition from -180.degree. to -90.degree. in phase, around its
pole frequency p, than is the phase characteristic of the transfer
function for conventional loop filters.
[0115] According to this second preferred embodiment of the
invention, it is contemplated that the hardware necessary for
realizing this three feedback path amplifier 30' is relatively
simple, especially because of the use of LC filter 32 within the
feedback loop. An exemplary realization of the loop filter portion
of amplifier 30' is illustrated in FIG. 17.
[0116] As shown in FIG. 17, the three feedback loops and adder 41
of amplifier 30' of FIG. 16 can be realized using operational
amplifier block 60 containing a single operational amplifier 68,
together with LC filter 32 itself. Output signal p(t) from power
stage 37 is applied to LC filter 32, which includes a series
inductor L and parallel capacitor C connected to ground, and which
is also affected by load resistor R.sub.L (i.e., the impedance of
the external load). The output of LC filter 32 is also fed back
through resistor network 62, to generate current i.sub.D at adder
node 41, which is at the virtual ground node at the inverting input
of operational amplifier 68. A pair of RC networks 64, 66 connect
output signal p(t) to adder node 41, generating feedback currents
i.sub.A, i.sub.C, respectively. Input signal i(t) is applied to
adder node 41 through input resistor R.sub.in. Operational
amplifier block 60 includes operational amplifier 68, which has its
non-inverting input at ground and its inverting input (i.e.,
virtual ground input) receiving summed current i.sub.B from adder
node 41, and feedback resistor R.sub.B1 and feedback capacitor
C.sub.B1.
[0117] The total loop transfer function H.sub.mae(s) of the circuit
of FIG. 17 is: 32 H mae ( s ) = - X ( s ) P ( s ) = [ H D ( s ) + H
A ( s ) + H C ( s ) ] H B ( s ) ( 27 )
[0118] where the component transfer functions H.sub.A(S),
H.sub.C(s), correspond to the transfer functions of blocks 64, 66,
respectively, where the component transfer function H.sub.D(s)
corresponds to the transfer function of LC filter 32 combined with
resistor block 62, and where the component transfer function
H.sub.B(S) corresponds to the transfer function of amplifier block
60 (and which thus multiplies the sum of the transfer functions
H.sub.A(S), H.sub.C(s), H.sub.D(S) summed at adder node 41).
[0119] Those skilled in the art will be readily able to choose
component values to effect the desired component transfer
functions. Specifically, component transfer function H.sub.A(S) can
be derived as: 33 H A ( s ) = I A ( s ) P ( s ) = R B1 sC B1 R B1 +
1 sC B1 = s R A1 R A2 R A1 + R A2 C A1 s R A1 R A2 R A1 + R A2 C A1
+ 1 R A1 + R A2 R A1 R A2 = sC A1 s R A1 R A2 R A1 + R A2 C A1 + 1
( 28 )
[0120] Component transfer function H.sub.C(s) can be derived as: 34
H C ( s ) = I C ( s ) P ( s ) = R C2 R C1 + R C2 1 R C3 + R C1 R C2
R C1 + R C2 s R C1 R C2 R C3 R C1 + R C2 + R C3 C B1 + 1 = R C2 ( R
C1 + R C2 ) R C3 + R C1 R C2 1 s R C1 R C2 R C3 R C1 + R C2 + R C3
C B1 + 1 ( 29 )
[0121] The combination of LC filter 32 and resistor block 62
derives component transfer function H.sub.D(S) as: 35 H D ( s ) = I
D ( s ) P ( s ) = 1 s 2 LC + s L R L + 1 R D1 + R D2 R D1 R D2 ( 30
)
[0122] The multiplicative transfer function H.sub.B(s) applied by
amplifier block 60 is: 36 H B ( s ) = X ( s ) I B ( s ) = - R B1 sC
B1 R B1 + 1 sC B1 = - R B1 sR B1 C B1 + 1 ( 31 )
[0123] It is contemplated that those skilled in the art having
reference to this specification can derive the overall transfer
function H.sub.mae(s) by substituting equations (28) through (31)
into equation (27). The error transfer function ETF.sub.mae(s) can
then be readily derived from: 37 ETF mae ( s ) = 1 1 + K H mae ( s
) ( 32 )
[0124] where gain K is the gain of the PWM modulator of comparator
35. For the example of FIG. 17, one can derive the signal transfer
function STF.sub.mae(s) as: 38 STF mae ( s ) = V out ( s ) V in ( s
) = K X ( s ) V in ( s ) H LC ( s ) 1 + K H mae ( s ) = K H B ( s )
H LC ( s ) R in 1 + K H mae ( s )
[0125] It is contemplated that those skilled in the art and having
reference to this specification will be readily able to optimize
amplifier 30', as realized according to FIG. 17 or otherwise, by
selection of the various components in the feedback loop circuit.
Specifically, referring to amplifier 30' illustrated in FIG. 16,
the parameters that can be used in connection with the optimization
are the load resistance R.sub.L, the inductor and capacitor values
in LC filter 32, the gains K.sub.a, K.sub.c, K.sub.d, and the poles
p.sub.a, p.sub.b, p.sub.c. Typically, the component values for LC
filter 32 are predetermined, and the load resistance R.sub.L will
have upper and lower bounds. According to one exemplary
implementation, the remaining six parameters (gains and poles) were
optimized using a constrained numerical optimization, such as can
be performed using conventional mathematics software (e.g., the
MATLAB computing environment available from The MathWorks, Inc.).
This exemplary optimization minimized the difference in dB between
the maximum and minimum gains of the signal transfer function
STF.sub.mae(s) at the highest frequency in the audio band of
interest, for load resistance R.sub.L at its upper and lower
bounds. This optimization reduced the amplitude response variation
due to variations in this load resistance R.sub.L. Additional
constraints in the optimization can include such parameters as
ripple stability (by setting a maximum transfer function amplitude
at the switching frequency F.sub.sw) closed-loop stability (by
setting minimum damping factor values), and distortion (by setting
an upper limit for open-loop distortion of the aliasing amplitude
error function G(y)). It is contemplated that those skilled in the
art having reference to this specification will be readily able to
effect this optimization.
[0126] According to the preferred embodiments of the invention,
therefore, important advantages can be readily attained. Primarily,
a class D amplifier can be constructed in which its loop filter
suppresses aliasing error in the base-band frequencies of interest.
The loop filter can be very efficiently implemented, and indeed can
be implemented using the output LC filter itself, which reduces the
complexity and also compensates for error introduced by the LC
filter, or by variations in the load impedance.
[0127] While the present invention has been described according to
its preferred embodiments, it is of course contemplated that
modifications of, and alternatives to, these embodiments, such
modifications and alternatives obtaining the advantages and
benefits of this invention, will be apparent to those of ordinary
skill in the art having reference to this specification and its
drawings. It is contemplated that such modifications and
alternatives are within the scope of this invention as subsequently
claimed herein.
* * * * *